Non-reciprocity of wave propagation is a fascinating property of a medium originating from time-reversal symmetry breaking According to the Casimir-Onsager principle, for a device to be non-reciprocal, its scattering matrix must depend on an odd vector upon time-reversal. For instance, in such a non-reciprocal device (e.g., isolator, diode), the waves are totally transmitted in one direction and perfectly reflected in the other. Recently, a few proposals for achieving unidirectional sound propagation in linear devices have been discussed, but most of these concepts use an asymmetric linear structure without any type of odd-vector bias, making the device totally symmetric upon time-reversal, and therefore completely reciprocal. These linear devices behave as asymmetrical mode converters, rather than as isolators. These linear devices cannot be used for sound isolation because if the input and output are reversed, as required in a device having the purpose of a diode between two ports, the propagation is strictly reciprocal.
A viable solution to achieve acoustic non-reciprocity, suitable for isolation, is to use non-linear media. For instance, one can pair a phononic crystal and a non-linear medium capable of converting the frequency of the wave. From one side, the wave is reflected because the crystal is operating in the band gap. From the other side, the wave frequency is converted into a value in the propagation band of the crystal, and therefore transmitted through the structure. However, this solution requires very high input powers and makes it difficult to efficiently operate with the low-intensity signals typically encountered in linear acoustics. As an additional drawback, particularly problematic for sound waves, it drastically modifies the frequency of the signal. In principle though, non-reciprocal propagation in linear systems is allowed by the laws of physics. Magnetic bias can induce non-reciprocity, like in the case of the acoustic Faraday effect, but magneto-acoustic effects are relatively weak and would require large devices considerably bigger than the wavelength. Mechanical motion has been proposed to realize an acoustic gyrator (a non-reciprocal phase shifter), but as in the case of magnetic bias, the obtained device is very bulky and stringently limited to transverse waves on pipes. A solution for a linear, compact acoustic non-reciprocal device for longitudinal waves in a gas (e.g., air) is still missing and highly desirable for audible sound isolation.